Geometric formula Square A side length C = 4as = A2 rectangle A and B side length C = 2 (A+B) S = AB triangle A, B, C- Half the height of S-H-A side perimeter A, B, C- internal angle where s = (a+b+c)/2s = ah/2 = ab/2sinc = [s (s-a) (s-b) (s-c)]1/2 = a2sinb sinc/(2sina) quadrilateral d, D. B side length H-A side height α-included angle S = ah = ABS in α rhombus A side length α-included angle D- long diagonal length D- short diagonal length S = DD/2 = A2Sinα trapezoid A and B- upper and lower bottom length H- height m- centerline length S = (a+b) h/ 2 = MH circle R- radius D- diameter C = 4 sector R- sector radius A- central angle C = 2r+2π s = π r2× (a/360) bow L- arc length B- chord length H- rise r-180-sin α) = r2arccos [(r-h)/r]-(r-h) (. The same angle or the complementary angle of the same angle is equal. 4. The same angle or the complementary angle of the same angle is equal. 5. There is only one straight line, and the known straight line is vertical. 6. Among all the line segments connected with points on a straight line, the shortest parallel axiom of a vertical line segment passes through a point outside the straight line. There is only one straight line parallel to this straight line. If both lines are parallel to the third line, the two lines are parallel to each other. The isosceles angles are equal and the two straight lines are parallel to each other. 10, the offset angles are equal, and the two straight lines are parallel to each other. 1 1 is complementary to the inner corner of the side, and the two straight lines are parallel to each other. 13, two straight lines are parallel. The internal dislocation angle is equal to 14, and the two straight lines are parallel. Theorem The sum of two sides of a triangle is greater than the third side 15. The difference between two sides of the reasoning triangle is less than the third side 17. Theorem The sum of three angles of a triangle is equal to 180 18. The two acute angles of a right triangle complement each other 19. The sum of the two angles of a triangle is equal to 18. The outer angle is equal to the sum of two non-adjacent inner angles. 20 Inference 3 An outer angle of a triangle is larger than the corresponding side of any inner angle that is not adjacent to it, 2 1 congruent triangles. The corresponding angles are equal. The 22-angle axiom has two triangles with equal angles. The 23-angle axiom has two angles and two equilateral triangles. It is inferred that there are two angles, and the opposite side of one angle corresponds to a triangle with two equal sides. The 25-sided axiom has three sides, corresponding to a triangle with two equal sides. 26 hypotenuse, right-angled side axiom has hypotenuse and right-angled side corresponding to two equal right-angled triangles. Theorem 1 The distances between points on the angular bisector are equal. Theorem 2 Reach the point where both sides of an angle are equidistant. On the bisector of this angle, the bisector of 29 angles is the set of all points with equal distance to both sides of the angle. The property theorem of isosceles triangle 30 The two base angles of isosceles triangle are equal. 3 1 Inference 1 The bisector of the top angle of the isosceles triangle bisects the bottom and is perpendicular to the bottom 32. The heights of the center line and the base coincide with each other. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60 34 isosceles triangle. Theorem: If the two angles of a triangle are equal, then the opposite sides of the two angles are equal (equilateral) 35 Inference 1 A triangle with three equal angles is an equilateral triangle 36 Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle 37 in a right triangle. If an acute angle is equal to 30, then the right-angled side it faces is equal to half of the hypotenuse. The median line of the hypotenuse of a right triangle is equal to half of the hypotenuse. Theorem 39 A point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment. The inverse theorem and the point where the two endpoints of a line segment are equal. On the midline of this line segment, the midline of line segment 4 1 can be regarded as a set of all points with equal distance from both ends of the line segment. Theorem 42: Two graphs that are symmetrical about a straight line are congruent. Theorem 43: Two figures are symmetrical about a straight line, then the symmetry axis is the median vertical line 44 Theorem 3: Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry. 45 Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line. 46 Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right triangle is equal to the square of hypotenuse C, that is, A+B = the inverse theorem of Pythagorean Theorem a+b=c 47. If the lengths of three sides of a triangle are related and a+b=c, then the triangle is a right triangle. Theorem 48 The sum of internal angles of quadrilateral is equal to 360 49, and the sum of internal angles of polygon is equal to 360 50. Theorem n The sum of the internal angles of a polygon is equal to (n-2) × 180 5 1. It is inferred that the sum of external angles of any polygon is equal to 360 52, and the diagonal of parallelogram is equal to 53. Parallelogram property theorem 2 Parallelogram with equal opposite sides 54 Parallelogram property theorem 3 Parallelogram diagonal bisection 56 Parallelogram decision theorem 1 Two sets of parallelograms with equal diagonal are parallelograms 57 Parallelograms decision theorem 2 Two sets of parallelograms with equal opposite sides are parallelograms 58 parallel sides. Shape Decision Theorem 3 The quadrilateral whose diagonal is bisected is a parallelogram 59. Parallelogram Decision Theorem 4 A group of parallelograms whose opposite sides are parallel and equal is a parallelogram 60. Rectangular property theorem 1 rectangular property theorem 6 1 rectangular property theorem 2. Rectangular diagonal is equal to 62. Rectangular decision theorem 1 a quadrilateral with three right angles is a rectangle 63. Theorem 2 A parallelogram with equal diagonals is a rectangle 64. The four sides of the diamond 1 are equal. Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines 66. Diamond area = half of diagonal product. That is, s=(a×b)÷2 67 rhombus decision theorem 1 A quadrilateral with four equal sides is a rhombus 68 rhombus decision theorem 2 A parallelogram with diagonal lines perpendicular to each other is a rhombus 69 square property theorem 1 All four corners of a square are right angles, and all four sides are equal to 70 square property theorem 2 Two diagonal lines of a square are equal and divided vertically. Each diagonal bisects a set of diagonals 7 1 theorem 1 congruence of two figures symmetrical about the center 72 Theorem 2 For two figures symmetrical about the center, the straight line of the symmetrical point passes through the symmetrical center and is bisected by the symmetrical center 73 Inverse Theorem If the straight line of the corresponding point of two figures passes through a point and is bisected by the point, the two figures are symmetrical about the point. Property theorem of isosceles trapezoid. The two angles of an isosceles trapezoid on the same base are equal. The two diagonals of an isosceles trapezoid are equal. 76 isosceles trapeziums have equal angles on the same base, which is an isosceles trapezoid. The diagonal trapezoid is an isosceles trapezoid. Theorem of bisecting line segments by 78 parallel lines. If a set of parallel lines cut on a straight line are equal, then the line segments cut on other straight lines are also equal. 79 Inference 1 Through a straight line parallel to the bottom of the trapezoid, the other waist 80 must be equally divided. Inference 2 Inference 2 Through a straight line parallel to the other side of the triangle, the third side must be bisected. 8 1 The midline theorem of the triangle is parallel to the third side. And equal to half of it. The trapezium midline theorem is parallel to the two bottoms and is equal to half the sum of the two bottoms. The basic properties of l=(a+b)÷2 s=l×h 83 (1) If a:b=c:d, then ad=bc. If ad=bc, then a:b=c:d 84 (2) Proportional property If A/B = C/D, then (A B)/B = (C D)/D 85 (3) Isometric property If A/B = C/D = ... = M/N (where (A+C) It is inferred that a straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segment obtained is proportional to Theorem 88. Then this straight line is parallel to the third side 89 of the triangle, parallel to one side of the triangle, and intersects the other two sides. The three sides of the cut triangle are proportional to the three sides of the original triangle. Theorem 90 A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides). The triangle formed is similar to the original triangle 9 1 similar triangle judgment theorem 1. Two triangles are similar (asa) 92. The two right triangles divided by the height on the hypotenuse are similar to the original triangle 93. Decision theorem 2, the two sides are proportional and the included angle is equal. Decision Theorem 3. Two triangles are similar (sas) 94. Three sides are proportional. Two triangles are similar (SSS) Theorem 95 If the hypotenuse and a right-angled side of a right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar. Theorem 96: 1 similar triangles corresponds to a high proportion. The ratio of the corresponding median line to the bisector of the corresponding angle is equal to the similarity ratio of 97. Property theorem 2. The ratio of similar triangles perimeter is equal to similarity ratio 98. Property theorem 3. The ratio of similar triangles area is equal to the square of similarity ratio 99. The sine of any acute angle is equal to the cosine of the remaining angle 100. The tangent of any acute angle is equal to the cotangent of the other angles.